arXiv subjects

Sergii M. Torba

Publications and source records attributed to Sergii M. Torba.

At least 37 records · Page 2Linked to original sources

arXiv2017

Pricing double barrier options on homogeneous diffusions: a Neumann series of Bessel functions representation

This paper develops a novel analytically tractable Neumann series of Bessel functions representation for pricing (and hedging) European-style double barrier knock-out options, which can be applied to the whole class of one-dimensional time-homogeneous diffusions even for the cases where the corresponding transition density is not known. The proposed numerical method is shown to be efficient and simple to implement. To illustrate the flexibility and computational power of the algorithm, we develop an extended jump to default model that is able to capture several empirical regularities commonly observed in the literature.

q-fin.CP↗

arXiv2017

Spectral parameter power series for arbitrary order linear differential equations

Let $L$ be the $n$-th order linear differential operator $Ly = ϕ_0y^{(n)} + ϕ_1y^{(n-1)} + \cdots + ϕ_ny$ with variable coefficients. A representation is given for $n$ linearly independent solutions of $Ly=λr y$ as power series in $λ$, generalizing the SPPS (spectral parameter power series) solution which has been previously developed for $n=2$. The coefficient functions in these series are obtained by recursively iterating a simple integration process, begining with a solution system for $λ=0$. It is shown how to obtain such an initializing system working upwards from equations of lower order. The values of the successive derivatives of the power series solutions at the basepoint of integration are given, which provides a technique for numerical solution of $n$-th order initial value problems and spectral problems.

math.CA↗

arXiv2017

On construction of transmutation operators for perturbed Bessel equations

A representation for the kernel of the transmutation operator relating the perturbed Bessel equation with the unperturbed one is obtained in the form of a functional series with coefficients calculated by a recurrent integration procedure. New properties of the transmutation kernel are established. A new representation of the regular solution of the perturbed Bessel equation is given presenting a remarkable feature of uniform error bound with respect to the spectral parameter for partial sums of the series. A numerical illustration of application to the solution of Dirichlet spectral problems is presented.

math.CA↗

arXiv2017

Transmutation operators: construction and applications

Recent results on the construction and applications of the transmutation (transformation) operators are discussed. Three new representations for solutions of the one-dimensional Schrödinger equation are considered. Due to the fact that they are obtained with the aid of the transmutation operator all the representations possess an important for practice feature. The accuracy of the approximate solution is independent of the real part of the spectral parameter. This makes the representations especially useful in problems requiring computation of large sets of eigendata with a nondeteriorating accuracy. Applications of the exact representations for the transmutation operators to partial differential equations are discussed as well. In particular, it is shown how the methods based on complete families of solutions can be extended onto equations with variable coefficients.

math.CA↗

arXiv2017

Analytic approximation of solutions of parabolic partial differential equations with variable coefficients

A complete family of solutions for the one-dimensional reaction-diffusion equation \[ u_{xx}(x,t)-q(x)u(x,t) = u_t(x,t) \] with a coefficient $q$ depending on $x$ is constructed. The solutions represent the images of the heat polynomials under the action of a transmutation operator. Their use allows one to obtain an explicit solution of the noncharacteristic Cauchy problem for the considered equation with sufficiently regular Cauchy data as well as to solve numerically initial boundary value problems. In the paper the Dirichlet boundary conditions are considered however the proposed method can be easily extended onto other standard boundary conditions. The proposed numerical method is shown to reveal good accuracy.

math.AP↗

arXiv2017

Solution of parabolic free boundary problems using transmuted heat polynomials

A numerical method for free boundary problems for the equation \[ u_{xx}-q(x)u=u_t \] is proposed. The method is based on recent results from transmutation operators theory allowing one to construct efficiently a complete system of solutions for this equation generalizing the system of heat polynomials. The corresponding implementation algorithm is presented.

math.AP↗

arXiv2017

Asymptotics with respect to the spectral parameter and Neumann series of Bessel functions for solutions of the one-dimensional Schrödinger equation

A representation for a solution $u(ω,x)$ of the equation $-u"+q(x)u=ω^2 u$, satisfying the initial conditions $u(ω,0)=1$, $u'(ω,0)=iω$ is derived in the form \[ u(ω,x)=e^{iωx}\left( 1+\frac{u_1(x)}ω+ \frac{u_2(x)}{ω^2}\right) +\frac{e^{-iωx}u_3(x)}{ω^2}-\frac{1}{ω^2}\sum_{n=0}^{\infty} i^{n}α_n(x)j_n(ωx), \] where $u_m(x)$, $m=1,2,3$ are given in a closed form, $j_n$ stands for a spherical Bessel function of order $n$ and the coefficients $α_n$ are calculated by a recurrent integration procedure. The following estimate is proved $\vert u(ω,x) -u_N(ω,x)\vert \leq \frac{1}{\vert ω\vert^2}\varepsilon_N(x)\sqrt{\frac{\sinh(2\mathop{\rm Im}ω\,x)}{\mathop{\rm Im}ω}}$ for any $ω\in\mathbb{C}\backslash \{0\}$, where $u_N(ω,x)$ is an approximate solution given by truncating the series in the representation for $u(ω,x)$ and $\varepsilon_N(x)$ is a nonnegative function tending to zero for all $x$ belonging to a finite interval of interest. In particular, for $ω\in\mathbb{R}\backslash \{0\}$ the estimate has the form $\vert u(ω,x)-u_N(ω,x)\vert \leq \frac{1}{\vertω\vert^2}\varepsilon_N(x)$. A numerical illustration of application of the new representation for computing the solution $u(ω,x)$ on large sets of values of the spectral parameter $ω$ with an accuracy nondeteriorating (and even improving) when $ω\rightarrow \pm \infty$ is given.

math.CA↗

arXiv2017

A Neumann series of Bessel functions representation for solutions of Sturm-Liouville equations

A Neumann series of Bessel functions (NSBF) representation for solutions of Sturm-Liouville equations and for their derivatives is obtained. The representation possesses an attractive feature for applications: for all real values of the spectral parameter $ω$ the difference between the exact solution and the approximate one (the truncated NSBF) depends on $N$ (the truncation parameter) and the coefficients of the equation and does not depend on $ω$. A similar result is valid when $ω\in\mathbb{C}$ belongs to a strip $|Imω|<C$. This feature makes the NSBF representation especially useful for applications requiring computation of solutions for large intervals of $ω$. Error and decay rate estimates are obtained. An algorithm for solving initial value, boundary value or spectral problems for the Sturm-Liouville equation is developed and illustrated on a test problem.

math.CA↗

arXiv2017

Representation of solutions to the one-dimensional Schrödinger equation in terms of Neumann series of Bessel functions

A new representation of solutions to the equation $-y"+q(x)y=ω^2 y$ is obtained. For every $x$ the solution is represented as a Neumann series of Bessel functions depending on the spectral parameter $ω$. Due to the fact that the representation is obtained using the corresponding transmutation operator, a partial sum of the series approximates the solution uniformly with respect to $ω$ which makes it especially convenient for the approximate solution of spectral problems. The numerical method based on the proposed approach allows one to compute large sets of eigendata with a nondeteriorating accuracy.

math.CA↗

arXiv2016

A Neumann series of Bessel functions representation for solutions of perturbed Bessel equations

A new representation for a regular solution of the perturbed Bessel equation of the form $Lu=-u"+\left( \frac{l(l+1)}{x^2}+q(x)\right)u=ω^2u$ is obtained. The solution is represented as a Neumann series of Bessel functions uniformly convergent with respect to $ω$. For the coefficients of the series explicit direct formulas are obtained in terms of the systems of recursive integrals arising in the spectral parameter power series (SPPS) method, as well as convenient for numerical computation recurrent integration formulas. The result is based on application of several ideas from the classical transmutation (transformation) operator theory, recently discovered mapping properties of the transmutation operators involved and a Fourier-Legendre series expansion of the transmutation kernel. For convergence rate estimates, asymptotic formulas, a Paley-Wiener theorem and some results from constructive approximation theory were used. We show that the analytical representation obtained among other possible applications offers a simple and efficient numerical method able to compute large sets of eigendata with a nondeteriorating accuracy.

math.CA↗

arXiv2015

Analytic approximation of transmutation operators and related systems of functions

In arXiv:1306.2914 a method for approximate solution of Sturm-Liouville equations and related spectral problems was presented based on the construction of the Delsarte transmutation operators. The problem of numerical approximation of solutions was reduced to approximation of a primitive of the potential by a finite linear combination of certain specially constructed functions obtained from the generalized wave polynomials introduced in arXiv:1208.5984 and arXiv:1208.6166. The method allows one to compute both lower and higher eigendata with an extreme accuracy. Since the solution of the approximation problem is the main step in the application of the method, the properties of the system of functions involved are of primary interest. In arXiv:1306.2914 two basic properties were established: the completeness in appropriate functional spaces and the linear independence. In this paper we present a considerably more complete study of the systems of functions. We establish their relation with another linear differential second-order equation, find out certain operations (in a sense, generalized derivatives and antiderivatives) which allow us to generate the next such function from a previous one. We obtain the uniqueness of the coefficients of expansions in terms of such functions and a corresponding generalized Taylor theorem. We construct the invertible integral operators transforming powers of the independent variable into the functions under consideration and establish their commutation relations with differential operators. We present some error bounds for the solution of the approximation problem depending on the smoothness of the potential and show that these error bounds are close to optimal in order. Also, we provide a rigorous justification of the alternative formulation of the proposed method allowing one to make use of the known initial values of the solutions at an endpoint.

math.CA↗

arXiv2014

Liouville transformation, analytic approximation of transmutation operators and solution of spectral problems

A method for solving spectral problems for the Sturm-Liouville equation $(pv^{\prime})^{\prime}-qv+λrv=0$ based on the approximation of the Delsarte transmutation operators combined with the Liouville transformation is presented. The problem of numerical approximation of solutions and of eigendata is reduced to approximation of a pair of functions depending on the coefficients $p$, $q$ and $r$ by a finite linear combination of certain specially constructed functions related to generalized wave polynomials introduced in arXiv:1208.5984, arXiv:1208.6166. The method allows one to compute both lower and higher eigendata with an extreme accuracy. Several necessary results concerning the action of the Liouville transformation on formal powers arising in the method of spectral parameter power series are obtained as well as the transmutation operator for the Sturm-Liouville operator $\frac{1}{r}\left(\frac{d}{dx}p\frac{d}{dx}-q\right)$.

math.CA↗

arXiv2014

Modulated electromagnetic fields in inhomogeneous media, hyperbolic pseudoanalytic functions and transmutations

The time-dependent Maxwell system describing electromagnetic wave propagation in inhomogeneous isotropic media in the one-dimensional case reduces to a Vekua-type equation for bicomplex-valued functions of a hyperbolic variable (see arXiv:1001.0552). Using this relation we solve the problem of the transmission through an inhomogeneous layer of a normally incident electromagnetic time-dependent plane wave. The solution is written in terms of a pair of Darboux-associated transmutation operators (see arXiv:1111.4449), and combined with the recent results on their construction (see arXiv:1208.6166, arXiv:1306.2914) can be used for efficient computation of the transmitted modulated signals. We develop the corresponding numerical method and illustrate its performance with examples.

math.AP↗

arXiv2014

Analysis of graded-index optical fibers by the spectral parameter power series method

Spectral parameter power series (SPPS) method is a recently introduced technique for solving linear differential equations and related spectral problems. In the present work we develop an approach based on the SPPS for analysis of graded-index optical fibers. The characteristic equation of the eigenvalue problem for calculation of guided modes is obtained in an analytical form in terms of SPPS. Truncation of the series and consideration in this way of the approximate characteristic equation gives us a simple and efficient numerical method for solving the problem. Comparison with the results obtained by other available techniques reveals clear advantages of the SPPS approach, in particular, with regards to accuracy. Based on the solution of the eigenvalue problem, parameters describing the dispersion are analyzed as well.

physics.optics↗

arXiv2014

Analytic approximation of transmutation operators and applications to highly accurate solution of spectral problems

A method for approximate solution of spectral problems for Sturm-Liouville equations based on the construction of the Delsarte transmutation operators is presented. In fact the problem of numerical approximation of solutions and eigenvalues is reduced to approximation of a primitive of the potential by a finite linear combination of generalized wave polynomials introduced in arXiv:1208.5984, arXiv:1208.6166. The method allows one to compute both lower and higher eigendata with an extreme accuracy.

math.CA↗

arXiv2014

Modified spectral parameter power series representations for solutions of Sturm-Liouville equations and their applications

Spectral parameter power series (SPPS) representations for solutions of Sturm-Liouville equations proved to be an efficient practical tool for solving corresponding spectral and scattering problems. They are based on a computation of recursive integrals, sometimes called formal powers. In this paper new relations between the formal powers are presented which considerably improve and extend the application of the SPPS method. For example, originally the SPPS method at a first step required to construct a nonvanishing (in general, a complex-valued) particular solution corresponding to the zero-value of the spectral parameter. The obtained relations remove this limitation. Additionally, equations with "nasty" Sturm-Liouville coefficients $1/p$ or $r$ can be solved by the SPPS method. We develop the SPPS representations for solutions of Sturm-Liouville equations of the form $$ (p(x)u')'+q(x)u=\sum_{k=1}^N λ^k R_k[u], \quad x\in(a,b)$$ where $R_k[u] :=r_k(x)u+s_k(x)u'$, $k=1,\ldots N$, the complex-valued functions $p$, $q$, $r_k$, $s_k$ are continuous on the finite segment $[a,b]$. Several numerical examples illustrate the efficiency of the method and its wide applicability.

math.CA↗

arXiv2014

Spectral parameter power series for polynomial pencils of Sturm-Liouville operators and Zakharov-Shabat systems

A spectral parameter power series (SPPS) representation for solutions of Sturm-Liouville equations of the form $$(pu')'+qu=u\sum_{k=1}^{N}λ^{k}r_{k}$$ is obtained. It allows one to write a general solution of the equation as a power series in terms of the spectral parameter $λ$. The coefficients of the series are given in terms of recursive integrals involving a particular solution of the equation $(pu_{0}')'+qu_{0}=0$. The convenient form of the solution provides an efficient numerical method for solving corresponding initial value, boundary value and spectral problems. A special case of the considered Sturm-Liouville equation arises in relation with the Zakharov-Shabat system. We derive an SPPS representation for its general solution and consider other applications as the one-dimensional Dirac system and the equation describing a damped string. Several numerical examples illustrate the efficiency and the accuracy of the numerical method based on the SPPS representations which besides its natural advantages like the simplicity in implementation and accuracy is applicable to the problems admitting complex coefficients, spectral parameter dependent boundary conditions and complex spectrum.

math.CA↗

arXiv2013

Construction of transmutation operators and hyperbolic pseudoanalytic functions

A representation for integral kernels of Delsarte transmutation operators is obtained in the form of a functional series with the exact formulas for the terms of the series. It is based on the application of hyperbolic pseudoanalytic function theory and recent results on mapping properties of the transmutation operators. The kernel $K_1$ of the transmutation operator relating $A=-\frac{d^2}{dx^2}+q_1(x)$ and $B=-\frac{d^2}{dx^2}$ results to be one of the complex components of a bicomplex-valued hyperbolic pseudoanalytic function satisfying a Vekua-type hyperbolic equation of a special form. The other component of the pseudoanalytic function is the kernel of the transmutation operator relating $C=-\frac{d^2}{dx^2}+q_2(x)$ and $B$ where $q_2$ is obtained from $q_1$ by a Darboux transformation. We prove the expansion theorem and a Runge-type theorem for this special hyperbolic Vekua equation and using several known results from hyperbolic pseudoanalytic function theory together with the recently discovered mapping properties of the transmutation operators obtain the new representation for their kernels. Several examples are given. Moreover, based on the presented results approaches for numerical computation of the transmutation kernels and for numerical solution of spectral problems are proposed.

math.AP↗